Problem: A circle with area $100\pi$ has a sector with a $\dfrac{4}{15}\pi$ radian central angle. What is the area of the sector? ${100\pi}$ $\color{#9D38BD}{\dfrac{4}{15}\pi}$ ${\dfrac{40}{3}\pi}$
The ratio between the sector's central angle $\theta$ and $2 \pi$ radians is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{2 \pi} = \dfrac{A_s}{A_c}$ $\dfrac{4}{15}\pi \div 2 \pi = \dfrac{A_s}{100\pi}$ $\dfrac{2}{15} = \dfrac{A_s}{100\pi}$ $\dfrac{2}{15} \times 100\pi = A_s$ $\dfrac{40}{3}\pi = A_s$